**2. Rheology**

Rheology describes the deformation of a body under the influence of stress. The nature of the deformation depends on the body's material conditions (Goodwin & Hughes, 2000). Ideal solids deform elastically, which means that the solid will deform and then return to its previous state once the force ceases. In this case, the energy needed for deformation will mainly be recovered after the stress terminates. If the same force is applied to ideal fluids, it will make them flow and the energy utilized will disperse within the fluid as heat. Thus, the energy will not be recovered once the forcing stress is terminated (Goodwin & Hughes, 2000).

For fluids a flow curve or rheogram is used to describe rheological properties. These properties may be of importance in anaerobic digestion for the dimensioning of e.g. feeding, pumping and stirring. Rheograms are constructed by plotting shear stress (τ) as a function of the shear rate () (Tixier *et al*., 2003; Guibad *et al*., 2005).

The stress applied to a body is defined as the force (F) divided by the area (A) over which this force is acting (Eq. 1). When forces are applied in opposite directions and parallel to the side of the body it is called shear stress (Goodwin & Hughes, 2000). Shear stress (τ*;* Pa*)* is one of the main parameters studied in rheology, since it is the force per unit area that a fluid requires to start flowing (Schramm, 2000). The shear rate (*;* s-1*)* describes the velocity gradient (Eq. 2). Hence, shear rate is the speed of a fluid inside the parallel plates generated when shear stress is applied (Pevere & Guibad, 2005).

$$\mathbf{r} = \mathbf{F}/\mathbf{A} = \mathbf{N}/\mathbf{m}^2 = \mathbf{P}\mathbf{a} \tag{1}$$

$$\gamma = \mathrm{d} \mathbf{v}\_{\times} / \mathrm{d} \mathbf{y} = (\mathrm{m}/\mathrm{s}) / \mathrm{m} \tag{2}$$

#### **2.1 Newtonian fluids**

Ideal fluids (e.g. water, methanol, olive oil and glycerol) perform linearly in rheograms, as illustrated for glycerol in figure 1, and are identified as Newtonian fluids. The Newtonian equation (Eq. 3) illustrates the flow behaviour of an ideal liquid (Schramm, 2000), where is the viscosity (Pa\*s). Dynamic viscosity, also called apparent viscosity, describes a fluid's resistance of deformation (Pevere & Guibad, 2005). In terms of rheology it is the relation of shear stress over the shear rate (Eq. 4). For Newtonian fluids the dynamic viscosity maintains a constant value meaning a linear relationship between and .

$$
\pi = \mathfrak{u} \,\,\,\,\gamma \tag{3}
$$

$$
\pi \eta = \pi / \gamma \tag{4}
$$

64 Biogas

solid concentration and on the characteristics of the organic material as well as on the interactions between particles and molecules in the solution (Foster, 2002). Therefore, this

The aim of this chapter is to briefly introduce the area of rheology and to present important parameters for rheological characterization of biogas reactor fluids. Examples are given from investigations on such parameters for lab-scale reactors digesting different substrates.

Rheology describes the deformation of a body under the influence of stress. The nature of the deformation depends on the body's material conditions (Goodwin & Hughes, 2000). Ideal solids deform elastically, which means that the solid will deform and then return to its previous state once the force ceases. In this case, the energy needed for deformation will mainly be recovered after the stress terminates. If the same force is applied to ideal fluids, it will make them flow and the energy utilized will disperse within the fluid as heat. Thus, the energy will not be recovered once the forcing stress is terminated (Goodwin & Hughes,

For fluids a flow curve or rheogram is used to describe rheological properties. These properties may be of importance in anaerobic digestion for the dimensioning of e.g. feeding, pumping and stirring. Rheograms are constructed by plotting shear stress (τ) as a function

The stress applied to a body is defined as the force (F) divided by the area (A) over which this force is acting (Eq. 1). When forces are applied in opposite directions and parallel to the side of the body it is called shear stress (Goodwin & Hughes, 2000). Shear stress (τ*;* Pa*)* is one of the main parameters studied in rheology, since it is the force per unit area that a fluid requires to start flowing (Schramm, 2000). The shear rate (*;* s-1*)* describes the velocity gradient (Eq. 2). Hence, shear rate is the speed of a fluid inside the parallel plates generated

Ideal fluids (e.g. water, methanol, olive oil and glycerol) perform linearly in rheograms, as illustrated for glycerol in figure 1, and are identified as Newtonian fluids. The Newtonian equation (Eq. 3) illustrates the flow behaviour of an ideal liquid (Schramm, 2000), where is the viscosity (Pa\*s). Dynamic viscosity, also called apparent viscosity, describes a fluid's resistance of deformation (Pevere & Guibad, 2005). In terms of rheology it is the relation of shear stress over the shear rate (Eq. 4). For Newtonian fluids the dynamic viscosity

maintains a constant value meaning a linear relationship between and .

τ = F/A = N/m2 = Pa (1)

= dvx/dy = (m/s)/m (2)

τ = \* (3)

= / (4)

of the shear rate () (Tixier *et al*., 2003; Guibad *et al*., 2005).

when shear stress is applied (Pevere & Guibad, 2005).

type of characterisation can be important in process monitoring and control.

**2. Rheology** 

2000).

**2.1 Newtonian fluids** 

When measuring the dynamic viscosity, the fluid is subjected to a force impact caused by moving a body in the fluid. Resistance to this movement provides a measure of fluid viscosity. The dynamic viscosity can be measured using a rotation rheometer. The device consists of an external fixed cylinder with known radius and an internal cylinder or spindle with known radius and height. The space between the two cylinders is filled with the fluid subjected to dynamic viscosity analysis.

Fig. 1. Rheogram – flow curve of glycerol (▲) at 20 °C with a linear relationship between shear stress (; Pa) and shear rate (; s-1), representing a Newtonian liquid.

#### **2.2 Limit viscosity**

Limit viscosity (lim) corresponds to the viscosity of a fluid at the maximum dispersion of the aggregates under the effect of the shear rate (Tixier & Guibad, 2003). The limit viscosity is estimated through the rheogram, when the dynamic viscosity becomes linear and constant. This parameter has been shown to be of great value when studying the rheological characteristics of sludge, since it determines the level of influence of important factors such as the total solids fraction (TS; Lotito *et al*., 1997). TS (%) and volatile solids (VS, % of TS) are parameters measured in the biogas process in order to control the amount of solids that may be transformed to methane. Also, Pevere and Guibad (2005) reported that the limit viscosity was sensitive to the physicochemical characteristics of granular sludge, i.e. it was influenced by changes in the particle size or the zeta potential.

start flowing. One type of these, the Bingham plastic, requires the shear stress to exceed a minimum yield stress value in order to go from high viscosity to low viscosity. After this change a linear relationship between the shear stress and the shear rate will prevail (Ryan,

Dilatant fluids become thicker when agitated, i.e. the viscosity increases proportionally with the increase of the shear rate. Like for the pseudoplastic fluids the stress duration has no influence, i.e. when the material is disturbed or the structure destroyed it will not go back to its previous state. Some examples of shear thickening behaviour are honey, cement and

Thixotropic fluids are generally dispersions, which when they are at rest construct an intermolecular system of forces and turn the fluid into a solid, thus, increasing the viscosity. In order to overcome these forces and make the fluid turn into a liquid and which may flow, an external energy strong enough to break the binding forces is needed. Thus, as above a yield stress is needed. Once the structures are broken, the viscosity is reduced when stirred until it receives its lowest possible value for a constant shear rate (Schramm, 2000). In opposite to pseudoplastic and dilatant fluids, the viscosity of thixotrpic fluids is time dependent: once the stirring has ended and the fluid is at rest, the structure will be rebuilt. This will inform about the fluid possibilities of being reconstructed. Wastewater and sewage sludge can be examples of fluids with thixotropic behaviour (Seyssieq & Ferasse, 2003) as

There are several rheological mathematical models applied on rheograms in order to transform them to information on fluid rheological behaviour. For non-Newtonian fluids the

The Herschel Bulkley model is applied on fluids with a non linear behaviour and yield stress. It is considered as a precise model since its equation has three adjustable parameters, providing data (Pevere & Guibaud, 2006). The Herschel Bulkley model is expressed in

The consistency index parameter () gives an idea of the viscosity of the fluid. However, to be able to compare -values for different fluids they should have similar flow behaviour index (n). When the flow behaviour index is close to 1 the fluid´s behaviour tends to pass from a shear thinning to a shear thickening fluid. When n is above 1, the fluid acts as a shear thickening fluid. According to Seyssiecq and Ferasse (2003) equation 5 gives fluid behaviour

τ = 0 + \* n (5)

three models presented below are mostly applied (Seyssiecq & Ferasse, 2003).

2003). Examples of Bingham plastic liquids are blood and some sewage sludge's.

**2.5.3 Dilatant fluids** 

ceramic suspensions.

**2.5.4 Thixotropic fluids** 

well as paints and soap.

**2.6 Rheological mathematical models** 

equation 5, where 0 represents the yield stress.

**2.6.1 Herschel Bulkley model** 

information as follows:
